Let a1 = 1 and an = n(an-1 + 1 ) for n=2,3,.....
Define
P = (1+ 1/a1 )(1+1/a2)......(1+1/an )
The Limit P when n tends to infinity is?
Step 1. Work out the first few terms to get an idea of what is happening. You should find that the first few values of are 1, 4, 15, 64, 325, 1956, and that the corresponding value of P is . This simplifies to , which looks suspiciously close to e.
Now see if you can turn those observations into proofs. You need to show first that the product is equal to . Then you want to show that .
Hint for that last part: let . Show that , and deduce that .